Laplacian spectra of a class of small-world networks and their applications.

Hongxiao Liu,Yi Qi,Zhongzhi Zhang,Maxim Dolgushev

doi:10.1038/srep09024

Hongxiao Liu, Yi Qi + Show 2 more

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https://doi.org/10.1038/srep09024

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Journal: Scientific Reports	Publication Date: Mar 12, 2015
Citations: 33	License type: CC BY 4.0

Affiliation: Fudan University, University of Freiburg

Abstract

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. In this paper, we introduce a family of small-world networks, parameterized through a variable d controlling the scale of graph completeness or of network clustering. We study the Laplacian eigenvalues of these networks, which are determined through analytic recursive equations. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension. Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. Although the networks have the same number of nodes (beads) and edges (springs) as the dual Sierpinski gaskets, they display rather different dynamic behavior.

Highlights

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics
We consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer
We start with a brief introduction to a family of small-world networks (SWNs) Vdg characterized by two parameters d and g, where d stands for the number of nodes of complete graph and g for the current generation

Summary

Introduction

One of the most crucial domains of interdisciplinary research is the relationship between the dynamics and structural characteristics. This allows us to analyze the spectra in depth and to determine the corresponding spectral dimension Based on these results, we consider the networks in the framework of generalized Gaussian structures, whose physical behavior is exemplified on the relaxation dynamics and on the fluorescence depolarization under quasiresonant energy transfer. The works from last century had solved the Laplacian eigenvalues for considerable amount of famous networks, like dual Sierpinski gaskets (in 2 or higher dimensions)[15,16], dendrimers[17], and Vicsek fractals[18,19] Another type of model structures, which often arise in the complex systems or polymer networks, are the so-called small-world networks (SWNs)[22,23,24,25].

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